Let a, b, c be positive integers such that a is a multiple of c and a2 + b2 is not a multiple of c. Prove that b is not a multiple of c.

Q: Determine whether it is possible to express 1 as a liner combination of 1 = m· 44 + n· 91 ,. for…

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Q: Show that the square of any positive Integer is of the form 3m or 3m+1 for some integer m

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Q: Show that the square of any integer is of the form 3m or 3m+1 but not of form 3m+2.

A: Proof :-

Q: Let a and b be positive integers. If ax + by = c has an integer solution (x, y), it is necessarily…

A: Consider the given equation. Find the characteristic for c if a, b are positive integers and ( x, y…

Q: Let a and d be integers. Prove that if d|(8a+7)and d|(5a+4), then d|3

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Q: Show that if a positive integer is odd, then it's square is odd.

A: To show that if a positive integer is odd then its square is odd

Q: a. Find all pairs of integers such that one is a multiple of 7 and the other is a multiple of 11 and…

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Q: Prove that a b + 2c + 3d + c+ 2d + 3a Let a, b, c, d be positive real numbers. C d + 2a + 3b + d a +…

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Q: Prove that 1+1=2

A: To prove: 1 + 1 = 2

Q: Let Q(x, y) be the statement "x + y = x – y." If the domain for both variables consists of all…

A: Given that Q(x, y) be the statement                         x + y = x -y .

Q: Assume that c and d are particular integers. Is 6c + 4d ² + 3 odd? 2

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Q: a

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Q: 8. Let a, b and c be any positive real numbers. Prove that b² + c² a²+b² 2c a² + c² + 2b 2a > a+b+c.

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Q: at a, b and c be real numbers such that a #0 and ac+bc+c 4ac.

A: Given data is : a , b and c are real number such that a≠0 and ac+bc+c2<0 Show that b2>4ac

Q: at any positive odd integer is of the

A: Here we have to show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5,…

Q: Let a and b be integers so that (a, b) = 1. Prove that (a + b, a – b) divides 2.

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Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

Q: Any integer of the form 12k +11 is also of the form a) 3m + 1 b) 5m +2 4m + 3 6m + 1

A: Answer

Q: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

Q: Q11. Define Exhaustive Proof and Proof by Cases. Prove that the only consecutive positive integers…

A: The proof by cases and Exhaustive proof are using the cases or examples and proving the theorem.…

Q: Prove that the sum of any five consecutive positive integers is always a multiple of five.

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Q: Prove that A2=I.

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Q: Find the number of nonnegative integer solutions to x₁ + x₂ + x3 + x4 + x5 ≤ 23. Find the number of…

A: We have to find the number of non negative integer solution of Note: According to the guidelines we…

Q: 3. Prove by contradiction, that if x and y are odd integers, then there is no integers z such that…

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Q: 2. Show that for all positive real numbers a and b, if a < b then a² < b².

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Q: 2. Prove that the difference of an odd integer and an even integer is odd.

A: Prove that the difference of an odd integer and an even integer is odd

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…